EE301 – INTRO TO AC AND SINUSOIDS
Learning Objectives a. Compare AC and DC voltage and current sources as defined by voltage polarity, current direction and magnitude over time b. Define the basic sinusoidal wave equations and waveforms, and determine amplitude, peak to peak values, phase, period, frequency, and angular velocity c. Determine the instantaneous value of a sinusoidal waveform d. Graph sinusoidal wave equations as a function of time and angular velocity using degrees and radians e. Define effective / root mean squared values f. Define phase shift and determine phase differences between same frequency waveforms Alternating Current (AC) With the exception of short-term capacitor and inductor transients, all voltages and currents we have seen up to this point have been “DC”—i.e., fixed in magnitude. Now we shift our focus to “AC” voltage and current sources. AC sources (usually represented by lowercase e(t) or i(t)) have a sinusoidal waveform. For an AC voltage, for example, the voltage polarity changes every cycle.
On the other hand, for an AC current, the current changes direction each cycle with the source voltage.
Voltage and Current Conventions When e has a positive value, its actual polarity is the same as the reference polarity. When i has a positive value, its actual direction is the same as the reference arrow.
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Sinusoids Since our ac waveforms (voltages and currents) are sinusoidal, we need to have a ready familiarity with the equation for a sinusoid. The horizontal scale, referred to as the “time scale” can represent degrees or time.
Period and Frequency The period (T) is the time taken to complete one full cycle. The frequency (f) is the number of cycles per second. The unit of frequency is hertz (Hz). 1 Hz = 1 cycle per second. The period and the frequency are related by the formula: 1 f (Hz) T
Amplitude and Peak-to-Peak Value The amplitude (denoted Em or Epk ) of a sine wave is the distance from its average value to its peak.
Peak-to-peak voltage (denoted Epp ) is the difference between the minimum and maximum peak.
The Basic Sine Wave Equation The equation for a sinusoidal source is given
eE m sin( ) V
where Em is peak coil voltage and is the angular position.
The instantaneous value of the waveform can be determined by solving the equation for a specific value of
e(37 ) 10sin(37 ) V = 6.01 V
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Radians We usually the angular position as a function of time. For starters, we measure angular velocity (denoted ) in radians per second where the conversion for radians to degrees is: 2 radians = 360º The angular position is then expressed in terms of angular velocity and time as: = t (radians) and thus we can rewrite the equation for a sinusoidal source as
e (t) = Em sin t (V) Relationship between , T and f The formulas that relate angular velocity ( in radians per second), the period (T in seconds) and the frequency (f in Hz) are = 2 f (rad/s) 2 2 f (rad/s) T Thus, we can represent a sinusoidal waveform (such as a voltage) as a function of time in terms of angular velocity () e (t) = Em sin t (V) or in terms of the frequency (f) e (t) = Em sin 2 f t (V) or in terms of period (T) t et() E sin2 (V) m T The instantaneous value is the value of the voltage at a particular instant in time.
Phase Shifts Often e does not pass through zero at t = 0 sec, and we account for this by a phase shift () . If e is shifted left (Leading), then
e = Em sin ( t + )
If e is shifted right (Lagging), then
e = Em sin ( t - )
The angle by which the wave LEADS or LAGS the zero point can be calculated based upon the Δt t 10 s 360 360 36 T 100 s
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1 Example: For the waveform shown determine (a) the peak voltage (b) the peak-to-peak voltage (c) the period (d) the frequency
Solution:
2 Example: Determine the frequency of the waveform shown:
Solution:
3 Example. Suppose we have a sine wave with a frequency of 60 Hz. What angle (in degrees) does this sine wave pass through during a time span of 10 msec?
Solution:
4 Example: For the sinusoidal current shown: (a) Determine the waveform’s period (b) Determine the frequency (c) Determine the value of Im (d) Determine the value of IPP Solution:
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5 Example: A sine wave has a value of 50V at =150˚.
What is the value of Em? Solution:
6 Example: A sine wave has a frequency of 100 Hz and an instantaneous value of 100 V at 1.25 msec. What is the voltage at 2.5 msec? Solution:
7 Example: Determine the equation for the waveform shown, expressing the phase angle in degrees.
Solution:
Phase Difference We are often interested in the phase difference between two sinusoids.
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8 Example: What is the phase relationship between the voltage and current waveforms if:
it15sin t 60 vt10sin t 20 Solution:
9 Example: Write the equation for the voltage shown below: Solution:
10 Example: Determine the phase angle between the voltage and the current shown below:
Solution:
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Effective (RMS) Values Effective values tell us about a waveform’s ability to do work. An effective value is an equivalent dc value. It tells you how many volts or amps of dc that a time-varying waveform is equal to in terms of its ability to produce average power.
V VVm 0.707 rms2 m I I m 0.707I rms 2 m
Converting from effective value back to peak values we use:
II2 1.414 I meffrms
VVmeffrms2 1.414 V
RMS stands for root mean square as this describes the operation we perform to find the effective value. The terms RMS and effective are synonymous.
11 Example: The 120 V dc source shown below delivers 3.6 W to the load. Determine the peak values of the sinusoidal voltage and current ( Em and Im ) such that the ac source delivers the same power to the load.
Solution:
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